As you may know, a polymorphic function is a function that works for many different types. For instance,

length::[a]->Int

can calculate the length of any list, be it a string String = [Char] or a list of integers [Int]. The type variablea indicates that length accepts any element type. Other examples of polymorphic functions are

fst::(a,b)->asnd::(a,b)->bmap::(a->b)->[a]->[b]

Type variables always begin in lowercase whereas concrete types like Int or String always start with an uppercase letter, that's how we can tell them apart.

There is a more explicit way to indicate that a can be any type

length::foralla.[a]->Int

In other words, "for all types a, the function length takes a list of elements of type a and returns an integer". You should think of the old signature as an abbreviation for the new one with the forall[1]. That is, the compiler will internally insert any missing forall for you. Another example: the types signature for fst is really a shorthand for

fst::foralla.forallb.(a,b)->a

or equivalently

fst::forallab.(a,b)->a

Similarly, the type of map is really

map::forallab.(a->b)->[a]->[b]

The idea that something is applicable to every type or holds for everything is called universal quantification. In mathematical logic, the symbol ∀[2] (an upside-down A, read as "forall") is commonly used for that, it is called the universal quantifier.

With explicit forall, it now becomes possible to write functions that expect polymorphic arguments, like for instance

foo::(foralla.a->a)->(Char,Bool)foof=(f'c',fTrue)

Here, f is a polymorphic function, it can be applied to anything. In particular, foo can apply it to both the character 'c' and the boolean True.

It is not possible to write a function like foo in Haskell98, the type checker will complain that f may only be applied to values of either the type Char or the type Bool and reject the definition. The closest we could come to the type signature of foo would be

bar::(a->a)->(Char,Bool)

which is the same as

bar::foralla.((a->a)->(Char,Bool))

But this is very different from foo. The forall at the outermost level means that bar promises to work with any argument f as long as f has the shape a -> a for some type a unknown to bar. Contrast this with foo, where it's the argument f who promises to be of shape a -> a for all types a at the same time , and it's foo who makes use of that promise by choosing both a = Char and a = Bool.

Concerning nomenclature, simple polymorphic functions like bar are said to have a rank-1 type while the type foo is classified as rank-2 type. In general, a rank-n type is a function that has at least one rank-(n-1) argument but no arguments of even higher rank.

The theoretical basis for higher rank types is System F, also known as the second-order lambda calculus. We will detail it in the section System F in order to better understand the meaning of forall and its placement like in foo and bar.

Haskell98 is based on the Hindley-Milner type system, which is a restriction of System F and does not support forall and rank-2 types or types of even higher rank. You have to enable the RankNTypes[3] language extension to make use of the full power of System F.

But of course, there is a good reason that Haskell98 does not support higher rank types: type inference for the full System F is undecidable, the programmer would have to write down all type signatures. Thus, the early versions of Haskell have adopted the Hindley-Milner type system which only offers simple polymorphic function but enables complete type inference in return. Recent advances in research have reduced the burden of writing type signatures and made rank-n types practical in current Haskell compilers.

and mutable arrays. The type variable s represents the state that is being manipulated. But unlike IO, these stateful computations can be used in pure code. In particular, the function

runST::(foralls.STsa)->a

sets up the initial state, runs the computation, discards the state and returns the result. As you can see, it has a rank-2 type. Why?

The point is that mutable references should be local to one runST. For instance,

v=runST(newSTRef"abc")foo=runST(readSTRefv)

is wrong because a mutable reference created in the context of one runST is used again in a second runST. In other words, the result type a in (forall s. ST s a) -> a may not be a reference like STRef s String in the case of v. But the rank-2 type guarantees exactly that! Because the argument must be polymorphic in s, it has to return one and the same type a for all states s; the result a may not depend on the state. Thus, the unwanted code snippet above contains a type error and the compiler will reject it.

Section goal = enable reader to come up with free theorems. no need to prove them, intuition is enough.

free theorems for parametric polymorphism.

Notes

↑Note that the keyword forall is not part of the Haskell 98 standard, but any of the language extensions ScopedTypeVariables, Rank2Types or RankNTypes will enable it in the compiler. A future Haskell standard will incorporate one of these.

↑The UnicodeSyntax extension allows you to use the symbol ∀ instead of the forall keyword in your Haskell source code.